SynopsisAn approximate analytic expression for the translational friction coefficient of a toroid modeled a s a continuous shell of frictional elements is derived using the Kirkwood approximation. The accuracy of this expression was determined by comparing the friction coefficients predicted by it to those predicted,by extrapolated shell-model calculations using the modified Oseen tensor. T o show that these calculatiops do indeed yield the correct friction coefficients, actual translational friction coefficients were determined by observing settling rates of macroscopic model rings or toroids in a high-viscosity silicone fluid. Our conclusion is that the approximate expression yields friction coefficients that are about 1.5-3%) low for finite rings. For thin rings, a comparison is also made with the exact result of Yamakawa and Yamaki [J. Chem. Phys. 57,1572Phys. 57, (1972 58,2049(1973l for the translational friction of plane polygonal rings. This comparison shows that the approximate expression yields results which are low by 2-3% unless t h e rings are extremely thin, in which case the error is larger. I n the limit of a n infinitely thin ring the approximate expression reduces to the Kirkwood result [ J . Polym. Sci. 12, 1 (1954)], which is low by 8.3%. We discuss hriefly how this work may he useful in determining the structure of DNA compacted by various solvent-electrolyte systems and polyamines.

SynopsisIn connection with our goal of calculating by practical methods the frictional properties of biopolymers from surface shells composed of spheres, we have investigated by the method of reflections the low-Reynolds-number hydrodynamic interaction between two unequal-sized spheres in translation. Previous results, in which the velocities were used as independent variables and which have the form of truncated infinite power series, were substa2tially extended. By inversion of the power series, new power series with better convergence properties were obtained. Equivalence of these inverted power series with those previously reported based on the method of reflections, when forces are used as independent variables, was demonstrated, and the solutions were again substantially extended. Applying the Lagrange interpolation to data generated from exact theories for the hydrodynamic interaction between two spheres, it was demonstrated that the various forms of the method of reflections do not just give reasonable power series, but actually yield optimal ones. These findings constitute a unification of diverse approaches and show methods of interconversion of results. On the basis of the power series obtained, a set of new hydrodynamic interaction tensors for two unequal spheres were derived. While the new tensors described the case of two unequal spheres with considerably more accuracy than those previously reported, direct application of these tensors to objects composed of more than two spheres revealed some unexpected problems resulting from overcorrection in the fourth-order term. However, when the tensors were preaveraged over all orientations of the multisphere object, a formula for the scalar translational friction coefficient was obtained that outperformed all but the most involved earlier approaches. It thus constitutes an improved and practical solution to the problem of computing translational friction coefficients of objects describable by a surface shell of many spheres, such as proteins.